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Does anyone have a good ref that reviews the history of dividing by zero? The current way of dealing with it just seems so ad hoc and dissatisfying... there has

by nonbel 10y ago

Does anyone have a good ref that reviews the history of dividing by zero? The current way of dealing with it just seems so ad hoc and dissatisfying... there has to be a long controversial history but searching around I find very little that isn't repeating the same stuff.

tnecniv 10y ago
You mean that it's undefined? It actually makes perfect sense that it is left this way. Consider addition and subtraction as functions: f(x) = x + 42 g(x) = x - 42 Now, realize that these functions are inverses of each other. Each one undoes the operation of the former: f(g(x)) = 42 + x - 42 = x g(f(x)) = x + 42 - 42 = x If you change 42 to any other real number, this relationship still holds. This is because f and g are functions that relate a number uniquely to another number. The inverse then, is just flipping that relationship (7 -> 49 has the inverse 49 -> 7). You can define similar functions for multiplication and division, and things will work out in the same fashion until you try to replace 42 with 0. What happens then is that the multiplication relationship is no longer unique. For f(x) = 0 * x f(x) is 0 for any choice of x! This is a problem when we try to flip the relationship as before to invert the operation. Since f(7) = 0 and f(49) = 0, should g(0) be 7 or 49? Moreover, the fact that the right and and left hand limit of 1/x as x -> infinity does not converge means that there's no obvious way to "plug the hole" so to speak and define 1/0 as some value.
- hashhar 10y ago
  Well you gave the answer. This was the way my maths teacher taught us about why 0 is a curious number.
gizmo686 10y ago
I'm in the same boat as only finding repeating stuff; but for other's reference, here is the repeating stuff: The earliest known attempt at dealing with division by zero comes from Brahmagupta in 628 (in the Brāhmasphuṭasiddhānta). His definition is as follows (non division-by-zero axioms ommited): 1) A non-zero number divided by 0 is a fraction with zero as the denominator. 2) Zero divided by a non-zero number is a fraction with zero as the numerator and the finite quantity as the denominator 3) Zero divided by zero is zero.
daxelrod 10y ago
Zero: The Biography of a Dangerous Idea by Charles Seife is one of the most engaging books I've read about mathematics. In its Amazon reviews, I also see The Nothing That Is: A Natural History of Zero by Robert Kaplan recommended.
bikenaga 10y ago
I'm a mathematician and I'm not aware that there was a "long controversial history". (I'll ask a colleague who teaches history of math and see if she knows anything.) I have a lot of students who plan to teach, and I tell them that their students (and other people) will probably ask why "you can't divide by zero". It's important to explain what is meant by "can't". (Does it mean no one knows how to do it? Or that some mathematical authorities issued a decree? Of course, it's nothing like that. Those kinds of misunderstandings can arise because people don't understand how math is done by mathematicians, because it is done in ways that are different from the way we do things in everyday life.) Anyway, the explanation is simple and quick enough that I can do it in any class where I'm discussing number systems (e.g. linear algebra, number theory, abstract algebra). The "tl;dr" is that if you want to "divide by 0" you will have to give up something else, and none of the things you have to choose from are things you'd want to give up. In more detail, suppose you could "divide by 0". Division is defined as multiplying by the multiplicative inverse. (If you don't like that definition, you have to explain what you'll substitute as the definition of division - and note that, mathematicians want a definition that extends smoothly to "number systems" that may be very unfamiliar.) So saying you can divide by 0 is the same as saying that 0 has a multiplicative inverse - call it 0^(-1). By definition of multiplicative inverse, 0 * 0^(-1) = 1. On the other hand, in any reasonable number system (specifically, in any ring), 0 * x = 0 for any x. The proof is easy - it uses the definition of "0", the definition of additive inverse, the distributive law, and associativity of addition. (Try it!) Therefore, 0 * 0^(-1) = 0, so 0 = 1. If this doesn't seem enough of a contradiction, just note that it follows from this (and the definition of "1") that x = 0 for all x. So the only "number" in the whole world is 0. Well, that makes life simple, but not very interesting. So: If you want to "divide by 0", you're going to have to give up one of those algebraic axioms I mentioned. Which one would you give up? Associativity of addition? The distributive law? I think it's really important to explain (particularly to kids learning math) that math is not a bunch of arbitrary rules. "Not dividing by 0" is not an arbitrary rule - it's a matter of making a trade-off.
- gizmo686 10y ago
  >So: If you want to "divide by 0", you're going to have to give up one of those algebraic axioms I mentioned. Which one would you give up? Associativity of addition? The distributive law? Just to expand on this, mathmaticians have done this; in several ways. For example, the Projetivly Extended real line, and Riemann sphere add "∞" to the Real or Complex numbers respectivly, such that 1/0=∞. Note that 0/0 remains undefined, ∞=-∞, 0∞ is undefined, and ∞ + ∞ is undefined (I am probably missing other "oddities" of these constructions). There is also a more general way of defining division by 0, that avoid undefined instances: wheels. As bikenaga mentions, the standard definition of a "reasonable" number system that involves addition and multiplication is a ring. In general, division by anything is not defined because elements are not guaranteed to have multiplicative inverses [0]. For example, the integers form a ring, but 5/3 is not defined in the integers. If you add the following two properties to a ring, you get an integral domain: 1) Commutativity: xy = yx 2) if xy=0 then x=0 or y=0. Again, the integers are an example. For a non integral domain ring, consider the integers mod 4, where 2 * 2 = 0. Once you have an integral domain, there is a standard way of defining division by any non-0 element: fractions. Informally, we that x^(-1) is the fraction 1/x, and a/b = x/y iff ay=bx. Addition and multiplication of fractions are defined as you learned in grade school. [1] As you would expect, applying this approach to the integers gives you the rational numbers. More formally, we defined the fraction x/y as the ordered pair (x,y). To define a wheel, we modify the above construction slightly. Specifically, we say that a/b = x/y iff there exists an s,s' such that (sa,sb) = (s'x,s'y) or sa/sb = s'x/s'y [2]. Addition and multiplication remain unchanged, but we define a new operation for taking inverses: /(x,y) = (y,x). That is to say that, to take the "inverse" of an element, you swap the numerator and denominator. In this system, we define 0 = (0,1) = 0/1 and 1 = (1,1) = 1/1. Division by 0 is now a simple matter: 0/0 = (0,1)/(0,1) = (0,1)(1,0) = (0,0) Notice that, under this construction, (0,0) is not the zero element; (0,1) is. Further, the equation (0,0) + x = (0,1) has no solution. If you keep poking at this structure, I am sure that you can find other bad things that happen. [0] In fact, depending on who you ask, a ring is not even required to have a multiplicative identity (eg. 1). [1] This construction gives you a structure known as a field; which is, in my opinion, the point where most non-mathematicians would start consider the algebraic structure to be a reasonable number system. [2] Under this construction, we can also loosen the requirements of the underlying ring. Specifically, any commutative ring will do. We do not require that xy=0 implies x=0 or y=0.

by nonbel 10y ago

tnecniv 10y ago

You mean that it's undefined? It actually makes perfect sense that it is left this way. Consider addition and subtraction as functions: f(x) = x + 42 g(x) = x - 42 Now, realize that these functions are inverses of each other. Each one undoes the operation of the former: f(g(x)) = 42 + x - 42 = x g(f(x)) = x + 42 - 42 = x If you change 42 to any other real number, this relationship still holds. This is because f and g are functions that relate a number uniquely to another number. The inverse then, is just flipping that relationship (7 -> 49 has the inverse 49 -> 7). You can define similar functions for multiplication and division, and things will work out in the same fashion until you try to replace 42 with 0. What happens then is that the multiplication relationship is no longer unique. For f(x) = 0 * x f(x) is 0 for any choice of x! This is a problem when we try to flip the relationship as before to invert the operation. Since f(7) = 0 and f(49) = 0, should g(0) be 7 or 49? Moreover, the fact that the right and and left hand limit of 1/x as x -> infinity does not converge means that there's no obvious way to "plug the hole" so to speak and define 1/0 as some value.

hashhar 10y ago
Well you gave the answer. This was the way my maths teacher taught us about why 0 is a curious number.

gizmo686 10y ago

I'm in the same boat as only finding repeating stuff; but for other's reference, here is the repeating stuff: The earliest known attempt at dealing with division by zero comes from Brahmagupta in 628 (in the Brāhmasphuṭasiddhānta). His definition is as follows (non division-by-zero axioms ommited): 1) A non-zero number divided by 0 is a fraction with zero as the denominator. 2) Zero divided by a non-zero number is a fraction with zero as the numerator and the finite quantity as the denominator 3) Zero divided by zero is zero.

daxelrod 10y ago

Zero: The Biography of a Dangerous Idea by Charles Seife is one of the most engaging books I've read about mathematics. In its Amazon reviews, I also see The Nothing That Is: A Natural History of Zero by Robert Kaplan recommended.

bikenaga 10y ago

I'm a mathematician and I'm not aware that there was a "long controversial history". (I'll ask a colleague who teaches history of math and see if she knows anything.) I have a lot of students who plan to teach, and I tell them that their students (and other people) will probably ask why "you can't divide by zero". It's important to explain what is meant by "can't". (Does it mean no one knows how to do it? Or that some mathematical authorities issued a decree? Of course, it's nothing like that. Those kinds of misunderstandings can arise because people don't understand how math is done by mathematicians, because it is done in ways that are different from the way we do things in everyday life.) Anyway, the explanation is simple and quick enough that I can do it in any class where I'm discussing number systems (e.g. linear algebra, number theory, abstract algebra). The "tl;dr" is that if you want to "divide by 0" you will have to give up something else, and none of the things you have to choose from are things you'd want to give up. In more detail, suppose you could "divide by 0". Division is defined as multiplying by the multiplicative inverse. (If you don't like that definition, you have to explain what you'll substitute as the definition of division - and note that, mathematicians want a definition that extends smoothly to "number systems" that may be very unfamiliar.) So saying you can divide by 0 is the same as saying that 0 has a multiplicative inverse - call it 0^(-1). By definition of multiplicative inverse, 0 * 0^(-1) = 1. On the other hand, in any reasonable number system (specifically, in any ring), 0 * x = 0 for any x. The proof is easy - it uses the definition of "0", the definition of additive inverse, the distributive law, and associativity of addition. (Try it!) Therefore, 0 * 0^(-1) = 0, so 0 = 1. If this doesn't seem enough of a contradiction, just note that it follows from this (and the definition of "1") that x = 0 for all x. So the only "number" in the whole world is 0. Well, that makes life simple, but not very interesting. So: If you want to "divide by 0", you're going to have to give up one of those algebraic axioms I mentioned. Which one would you give up? Associativity of addition? The distributive law? I think it's really important to explain (particularly to kids learning math) that math is not a bunch of arbitrary rules. "Not dividing by 0" is not an arbitrary rule - it's a matter of making a trade-off.

gizmo686 10y ago
>So: If you want to "divide by 0", you're going to have to give up one of those algebraic axioms I mentioned. Which one would you give up? Associativity of addition? The distributive law? Just to expand on this, mathmaticians have done this; in several ways. For example, the Projetivly Extended real line, and Riemann sphere add "∞" to the Real or Complex numbers respectivly, such that 1/0=∞. Note that 0/0 remains undefined, ∞=-∞, 0∞ is undefined, and ∞ + ∞ is undefined (I am probably missing other "oddities" of these constructions). There is also a more general way of defining division by 0, that avoid undefined instances: wheels. As bikenaga mentions, the standard definition of a "reasonable" number system that involves addition and multiplication is a ring. In general, division by anything is not defined because elements are not guaranteed to have multiplicative inverses [0]. For example, the integers form a ring, but 5/3 is not defined in the integers. If you add the following two properties to a ring, you get an integral domain: 1) Commutativity: xy = yx 2) if xy=0 then x=0 or y=0. Again, the integers are an example. For a non integral domain ring, consider the integers mod 4, where 2 * 2 = 0. Once you have an integral domain, there is a standard way of defining division by any non-0 element: fractions. Informally, we that x^(-1) is the fraction 1/x, and a/b = x/y iff ay=bx. Addition and multiplication of fractions are defined as you learned in grade school. [1] As you would expect, applying this approach to the integers gives you the rational numbers. More formally, we defined the fraction x/y as the ordered pair (x,y). To define a wheel, we modify the above construction slightly. Specifically, we say that a/b = x/y iff there exists an s,s' such that (sa,sb) = (s'x,s'y) or sa/sb = s'x/s'y [2]. Addition and multiplication remain unchanged, but we define a new operation for taking inverses: /(x,y) = (y,x). That is to say that, to take the "inverse" of an element, you swap the numerator and denominator. In this system, we define 0 = (0,1) = 0/1 and 1 = (1,1) = 1/1. Division by 0 is now a simple matter: 0/0 = (0,1)/(0,1) = (0,1)(1,0) = (0,0) Notice that, under this construction, (0,0) is not the zero element; (0,1) is. Further, the equation (0,0) + x = (0,1) has no solution. If you keep poking at this structure, I am sure that you can find other bad things that happen. [0] In fact, depending on who you ask, a ring is not even required to have a multiplicative identity (eg. 1). [1] This construction gives you a structure known as a field; which is, in my opinion, the point where most non-mathematicians would start consider the algebraic structure to be a reasonable number system. [2] Under this construction, we can also loosen the requirements of the underlying ring. Specifically, any commutative ring will do. We do not require that xy=0 implies x=0 or y=0.